How do you graph #R(x) = (x + 2)/(x^2 - 4)#?

To graph the function R(x) = (x + 2)/(x^2 - 4), follow these steps:

1. Determine the domain of the function by finding the values of x for which the denominator (x^2 - 4) is equal to zero. In this case, the denominator factors as (x - 2)(x + 2), so the function is undefined at x = 2 and x = -2. Therefore, the domain is all real numbers except x = 2 and x = -2.

2. Find the vertical asymptotes by setting the denominator equal to zero and solving for x. In this case, x^2 - 4 = 0, which factors as (x - 2)(x + 2) = 0. So, the vertical asymptotes are x = 2 and x = -2.

3. Determine the horizontal asymptote by analyzing the behavior of the function as x approaches positive or negative infinity. In this case, as x approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote is y = 0.

4. Find the x-intercepts by setting the numerator equal to zero and solving for x. In this case, (x + 2) = 0, so x = -2. Therefore, the x-intercept is (-2, 0).

5. Find the y-intercept by evaluating the function at x = 0. In this case, R(0) = (0 + 2)/(0^2 - 4) = 2/(-4) = -1/2. Therefore, the y-intercept is (0, -1/2).

6. Plot the vertical asymptotes, horizontal asymptote, x-intercept, and y-intercept on the coordinate plane.

7. To determine the behavior of the function between the vertical asymptotes, choose some test points within each interval and evaluate the function at those points. Plot the resulting points on the graph.

8. Connect the plotted points smoothly to form the graph of R(x) = (x + 2)/(x^2 - 4).

Note: It may be helpful to use a graphing calculator or software to visualize the graph accurately.